Selected elevations and forbidden activations

Dynamic Resonant Tunneling

8. Selected elevations and forbidden activations

(36)

This occurs approximately at the time scale

(37)

or, for a given τ, the transition occurs for the following incoming energy:

(38)

Figure 3. Comparison between the logarithm of the exact numerical solution (solid line) and the analytical approxima‐

tion (dashed line), i.e., Eq. (33), for

λ₀=100,L U =150, Ω/^{U =0.6=Ω}T,τ=τ_T=280U.

Clearly, this energy is lower than the minimum resonance energy

(39)

In Figure 4, the dependence of the spectrum on the transition time-scale τ is presented for three different values: below τT where activation prevails, above τT when simple tunneling wins, and when they are equal, and the outgoing particle’s spectrum has two equally probable outgoing energies.

It should be stressed that, because these peaks are exponentially narrow, the transition is extremely abrupt (i.e., the process resembles a phase transition). The identification of the process as a phase transition was first suggested by Azbel [17].

However, it was wrongly assumed that, if the particle’s incoming energy matches the quasi- eigenstate energy, then an eigenstate-assisted activation (EAA) effect occurs (i.e., if Ω>Ωmin*, then activation will definitely increase). In fact, it will be shown that this process is more complicated, and at some energies (above Ωmin*), activation is “totally” suppressed.

Figure 4. The (logarithm of) the exit probability as a function of the activation energy ωact for the parameters λ₀=100, L U =150, Ω

/

^U=0.6, for three different perturbation time scales: τ = τT (solid line), τ = τT + 20 (dashed line) and τT − 20 (dotted line).

(40)

where P(ωactΩ) is the probability of an incoming particle with energy Ω to exit the barrier with the energy ωact.

Figure 5. Schematic illustration of the suppressed activation. For most energies, activation occurs (i.e., ωact ≅ U); how‐

ever, for the specific energies (i.e., Ω=Ωm), activation is suppressed and ωact ≅ Ω = Ωm.

Figure 6. Absolute value of the transmission coefficient a(ω) as a function of the activation energy ωact. The dashed curve corresponds to the case Ω/U = 0.6 and the solid line corresponds to Ω/U = 0.56. The other parameters are λ₀=100, L U=10, τU=60.6.

In Figure 7 the mean activation energy <ωact> is plotted as a function of the perturbation time scale τ, and in Figure 8, <ωact> is plotted as a function of the incoming particle’s energy Ω. It is clearly seen that activation (<ωact> ≅ U) occurs mainly below τ<τT. However, even below this time-scale, there are specific values of τ, for which activation is suppressed (i.e., <ωact> ≅ Ω).

Similarly, activation occurs <ωact> ≅ U mainly above Ω>ΩT; however, even in the activation regime, there are specific energies for which <ω_act> ≅ Ω (i.e., suppressed activation).

(40)

where P(ωactΩ) is the probability of an incoming particle with energy Ω to exit the barrier with the energy ωact.

Figure 5. Schematic illustration of the suppressed activation. For most energies, activation occurs (i.e., ωact ≅ U); how‐

ever, for the specific energies (i.e., Ω=Ωm), activation is suppressed and ωact ≅ Ω = Ωm.

Figure 6. Absolute value of the transmission coefficient a(ω) as a function of the activation energy ωact. The dashed curve corresponds to the case Ω/U = 0.6 and the solid line corresponds to Ω/U = 0.56. The other parameters are λ₀=100, L U=10, τU=60.6.

In Figure 7 the mean activation energy <ωact> is plotted as a function of the perturbation time scale τ, and in Figure 8, <ωact> is plotted as a function of the incoming particle’s energy Ω. It is clearly seen that activation (<ωact> ≅ U) occurs mainly below τ<τT. However, even below this time-scale, there are specific values of τ, for which activation is suppressed (i.e., <ωact> ≅ Ω).

Similarly, activation occurs <ωact> ≅ U mainly above Ω>ΩT; however, even in the activation regime, there are specific energies for which <ω_act> ≅ Ω (i.e., suppressed activation).

Figure 7. Mean activation energy as a function of the time-scale τ for the parameters: Ω/U = 0.6, L U =6,λ₀=100.

When τ<τT, two important things occur: (1) At two specific times, the particle’s incoming energy is equal to the eigenenergy of the quasi-bound state of the varying well. (2) The well varies quickly enough so the particle has no time to escape from the well.

As a consequence of these two, the particle’s state changes with the well’s eigenstate; therefore, it is easier to excite the particle energetically. That was the logic that led Azbel to predict the EAA effect. Indeed, this effect does occur, and it is clearly seen (see Figure 4) that, when τ<τT, then, for most values of τ, the spectrum’s energy is concentrated around the barrier’s height U. However, this process cannot last if the particle cannot dwell inside the quasi-bound state.

This event occurs when there is destruction interference inside the well.

Had it been a stationary eigenstate with an eigenenergy Ω0 the eigenstate would accumulate a linear phase [i.e., exp(−iΩ0t)].

Figure 8. Mean activation energy as a function of the incoming energy Ω/U for τU =79, L U =6,λ₀=100, Ω_R is the resonance energy

However, because the quasi-bound state evolves in time, it gains the integral

(41)

When the incoming energy is above the minimum eigenenergy (i.e., Ω>Ω_min*=U−λ₀²/4τ²), there are two times, in which Ω=Ω*(t1)=Ω*(t2) (see Figure 9), and due to the temporal symmetry of the perturbation t1=−t2. Therefore, the particle has two options to be temporally bounded to the quasi-eigenstate: it can either begin at t1 and gain the phase exp

(

⁻ⁱt

^∫

₁

t

dt 'Ω^*(t ')

)

^{or at t2 and}

gain the phase exp

(

^{−iΩ(t2−t1)−i}t

^∫

₂ t

dt 'Ω^*(t ')

)

. If the two components are out of phase and a destructive interference occurs [33], i.e.,

(42)

the particle cannot survive within the well, and activation is frustrated.

Figure 9. When the minimum of the resonance energy of the perturbation is lower than the incoming energy Ω, then the instantaneous resonance energy Ω*t crosses the incoming energy twice (at t1 and t2). For successful activation, the cumulative phase between these two events must be constructive.

In our case, at the vicinity of the parabola peak,

However, because the quasi-bound state evolves in time, it gains the integral

(41)

When the incoming energy is above the minimum eigenenergy (i.e., Ω>Ω_min*=U−λ₀²/4τ²), there are two times, in which Ω=Ω*(t1)=Ω*(t2) (see Figure 9), and due to the temporal symmetry of the perturbation t1=−t2. Therefore, the particle has two options to be temporally bounded to the quasi-eigenstate: it can either begin at t1 and gain the phase exp

(

⁻ⁱt

^∫

₁

t

dt 'Ω^*(t ')

)

^{or at t2 and}

gain the phase exp

(

^{−iΩ(t2−t1)−i}t

^∫

₂ t

dt 'Ω^*(t ')

)

. If the two components are out of phase and a destructive interference occurs [33], i.e.,

(42)

the particle cannot survive within the well, and activation is frustrated.

Figure 9. When the minimum of the resonance energy of the perturbation is lower than the incoming energy Ω, then the instantaneous resonance energy Ω*t crosses the incoming energy twice (at t1 and t2). For successful activation, the cumulative phase between these two events must be constructive.

In our case, at the vicinity of the parabola peak,

(43)

After substituting (43) into (42), the values of the forbidden energies Ωm, for which destructive interference occurs and the activation is suppressed, are directly given

(44)

In each one of these energies, the activation is suppressed.

To determine these energies more accurately, we take advantage of the fact that, at the vicinity of the minimum Ωmin*, the instantaneous resonance energy has a parabola shape; therefore, any varying potential with the same parabola should have approximately the same suppressed energies. Therefore, we replace the Gaussian with a parabolic function, that is, we choose Eq.

(28) for the perturbation, namely, f(t/τ) ≅ (λ₀/τ)(1 − t²/τ²), then

(45)

Therefore, the integral equation

(46)

reduces to the differential equation

(47)

where we used the dimensionless parameters n=(ω−Ω)τ and .

After linearization of the Green function, Eq. (47) can be approximated to

(48)

where again K≡ U−Ω.

The solution that maintain the boundary conditions that s(n → ∞) → 0 is

(49)

where ξ≡(n + K(λ0/τ − 2K)τ)/(2λ0K)^1/3, ξ0≡(K(λ0/τ − 2K)τ)/(2λ0K)^1/3 and Ai and Bi are the Airy functions [40].

Therefore, it is clear that activation is suppressed when

(50)

which, in the slowly varying approximation (i.e., large τ), correspond to (see [40])

(51)

Therefore, the incoming energies for which Ωact ≅ Ω, and thus no activation occurs, are approximately (see Figures 10 and 11)

(52)

Therefore, Eq. (42) should be rewritten more accurately as

(53)

like destructive interference condition in the WKB approximation (see, for example, ref. [19]).

Eq. (53) can be applied to any varying potential whose temporal shape has minima.

(49)

where ξ≡(n + K(λ0/τ − 2K)τ)/(2λ0K)^1/3, ξ0≡(K(λ0/τ − 2K)τ)/(2λ0K)^1/3 and Ai and Bi are the Airy functions [40].

Therefore, it is clear that activation is suppressed when

(50)

which, in the slowly varying approximation (i.e., large τ), correspond to (see [40])

(51)

Therefore, the incoming energies for which Ωact ≅ Ω, and thus no activation occurs, are approximately (see Figures 10 and 11)

(52)

Therefore, Eq. (42) should be rewritten more accurately as

(53)

like destructive interference condition in the WKB approximation (see, for example, ref. [19]).

Eq. (53) can be applied to any varying potential whose temporal shape has minima.

Figure 10. Approximate analytical expression of the forbidden time-scales (dashed curves) with the exact numerical solution of the probability density (the darker the spot the higher probability it represents). τm are the time scales for suppressed activations (52). τR is the minimum time scale τ_R≡−λ₀G_Ω⁺(0), and τT is the transition time when activation wins. With the parameters: Ω

/

^{U =0.6,λ}0=100, and L U =10.

Figure 11. Presentation of the analytical suppressed activation energies Ωm (dashed lines) on top of the numerical solu‐

tion of the probability to be activated to energy ωact (the darker the spot, the higher the probability it represents). ΩT is the transition energy when activation wins, and ΩR is the minimum resonant energy. The other parameters are τU =75,λ₀=100and L U =10.

Because (52) was derived for any potential, which can be approximated by a parabola [Eq. (28)], then the same conclusions and the same suppression of activation are valid to periodic potentials of the type

(54)

See ref. [23] for a more extensive study on these potentials; nevertheless, the suppressed activation energies (52) and (53) are still valid for these potentials.